You can use the Detach Attach technic to migrate site collection from a 2010 farm to a 2013 farm following these steps :
Detach Attach High level steps
Detach Attach Detailed steps
- Find out your site collection content DB name.
- Plan for capacity.
- Clean up before you upgrade
- Detach the content DB from the source farm containing your site collection( important when you have active workflows ).
- backup the content DB.
- Reattach the contend DB to the same source farm restoring the site to its initial state.
- restore the content DB from the backup to the target SQL server ( done by DBA ), make sure permission for the restored DB matches other content DB on the SQL server, so the SP_Farm account for example have to have DB creator on this DB.
- Test-SPcontentdatabase run PowerShell command to test the DB and produce pre mounting report
- Examine the report and resolve errors and warning by copying customizations (web-parts, features ) to the new farm.
- Mount-SPcontentdatabase Mount the Content Db using PowerShell
- Upgrade site collection to the new look and feel of SharePoint 2013
- the target 2013 farm is already installed
- web application already setup
- *copy service application database from the 2010 to 2013
Overview of the upgrade process from SharePoint 2010 to SharePoint 2013
Upgrading from SharePoint 2010 to SharePoint 2013 Step by Step